I want to apply an additional texture to my cup on the faces selected in the top screenshot. But the Normal UV map currently takes up all space
You're almost there. Here's how you do it.
First add a new material: Select your object and in materials settings press the +, then press New and name your material.

Next create a new UV vertex group, and unwrap it: Switch to UV Edit mode. Select the vertices or faces you want to add your new material too. Add a new UV Map and name it, then U unwrap your selected vertices, make sure you have your new UV Map selected in the active UV layer in the upper right corner of the image editor window.

Assign your texture: Back to Layout mode. In Edit Mode. Keep your vertices or faces selected, make sure the new material is selected in the Materials settings window, click Assign. Then you have to add the image texture. Go to the Shader Editor. Shift+A to add nodes. Add an Image Texture node and a UVMap node, set it up like in the image below. Depending on the look you're after, you may want to adjust some settings, like Transmission and Roughness.

Yours will look better than this with some fine adjustments.